I can't find much information about this flare gun beyond that it's of French manufacture and has a percussion cap ignition system, thus dating it to probably the Nineteenth Century. The cup at the end, I gather, is used for the flaring material. But I'm at a loss to figure out how the flare would be propelled.

All guns in your link are either metal, or some substrate covered by metal. The body and barrel of the gun in this entry is wood, except for the mouth and interior tube. The interior seems to be lined with a metal tube, maybe brass. The left side of the metal lined and reinforced bowl at the end of the gun appears to have a notch.

I think the picture is of the gun open for loading. Most likely the flare and primer is loaded, the gun closed, then held over the head - trigger pulled. The signal is ejected from the left side notch (POV of shooter) of the barrel.

There were similar flare guns used during the Civil War. They did not shoot a projectile as modern flare guns do today. A preloaded cartridge containing compounds that would burn in different colors or sequences of colors. The user selected the appropriate cartridge, loaded it into the gun, held it high overhead and pulled the trigger. It was essentially a pyrotechnic version of a flashlight.There are a few Civil War examples here: http://armscollectors.com/mgs/flare_pistols.htm